FINITE MIXTURE DISTRIBUTIONS

Transcription

1 MONOGRAPHS ON APPLl~[) PROBABILITY AND STATISTICS FINITE MIXTURE DISTRIBUTIONS

2 MONOGRAPHS ON APPLIED PROBABILITY AND STATISTICS General Editor D.R. COX, FRS Also available in the series Probability, Statistics and Time M.S. Bartlett The Statistical Analysis of Spatial Pattern M.S. Bartlett Stochastic Population Models in Ecology and Epidemiology M.S. Bartlett Risk Theory R.E. Beard, T. Pentikilinen and E. Pesonen Point Processes O.R. Cox and V. Isham Analysis of Binary Data D.R. Cox The Statistical Analysis of Series of Events D.R. Cox and P.A.W. Lewis Queues O.R. Cox and W.L. Smith Stochastic Abundance Models E. Engen The Analysis of Contingency Tables B.S. Everitt Finite Mixture Distributions B.S. Everitt and O.J. Hand Population Genetics W.J. Ewens

3 Classification A.D. Gordon Monte Carlo Methods J.M. Hammersley and D.C. Handscomb Identification of Outliers D.M. Hawkins Multivariate Analysis in Behavioural Research A.E. Maxwell Applications of Queueing Theory G.F. Newell Some Basic Theoryfor Statistical Inference E.J.G. Pitman Statistical Inference S.D. Silvey Models in Regression and Related Topics P. Sprent Sequential Methods in Statistics G.B. Wetherill

4 Finite Mixture Distributions B. S. EVERITT Head of Biometrics Unit, Institute of Psychiatry and DJ. HAND Lecturer, Biometrics Unit, Institute of Psychiatry LONDON NEW YORK CHAPMAN AND HALL

5 First published in 1981 by Chapman and Hall Ltd II New Fetter Lane, London EC4P 4EE Published in the USA by Chapman and Hall in association with Methuen, Inc. 733 Third Avenue, New York. NY B.S. Everitt and 0.1. Hand Softcover reprint of the hardcover 1st edition 1981 ISBN-13: All rights reserved. No part of this book may be reprinted, or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage and retrieval system, without permission in writing from the Publisher. British Library Cataloguing in Publication Data Everitt. B S Finite mixture distributions. - (Monographs on applied probability and statistics). I. Distribution (Probability theory) I. Title II. Series 5195'32 QA ISBN-13: e-isbn-13: : /

6 Contents Preface page xi 1 General introduction Introduction Some applications of finite mixture distributions Definition Estimation methods Maximum likelihood Bayesian estimation Inversion and error minimization Other methods Estimating the number of components Summary 23 2 Mixtures of normal distributions Introduction Some descriptive properties of mixtures of normal distributions Estimating the parameters in normal mixture distributions Method of moments estimation Maximum likelihood estimation Maximum likelihood estimates for grouped data Obtaining initial parameter values for the maximum likelihood estimation algorithms Graphical estimation techniques Other estimation methods Summary 57 3 Mixtures of exponential and other continuous distributions Exponential mixtures 59

7 VIll CONTENTS 3.2 Estimating exponential mixture parameters The method of moments and generalizations Maximum likelihood Properties of exponential mixtures Other continuous distributions Non-central chi-squared distribution Non-central F distribution Beta distributions Doubly non-central t distribution Planck's distribution Logistic Laplace Weibull Gamma Mixtures of different component types Summary 87 4 Mixtures of discrete distributions Introduction Mixtures of binomial distributions Moment estimators for binomial mixtures Maximum likelihood estimators for mixtures of binomial distributions Other estimation methods for mixtures of binomial distributions Mixtures of Poisson distributions Moment estimators for mixtures of Poisson distributions Maximum likelihood estimators for a Poisson mixture Mixtures of Poisson and binomial distributions Mixtures of other discrete distributions Summary Miscellaneous topics Introduction \ Determining the number of components in a mixture Informal diagnostic tools for the detection of mixtures \ Testing hypotheses on the number of components in a mixture 116

8 CONTENTS 5.3 Probability density function estimation 5.4 Miscellaneous problems 5.5 Summary References Index ix

9 Preface Finite mixture distributions arise in a variety of applications ranging from the length distribution of fish to the content of DNA in the nuclei of liver cells. The literature surrounding them is large and goes back to the end of the last century when Karl Pearson published his well-known paper on estimating the five parameters in a mixture of two normal distributions. In this text we attempt to review this literature and in addition indicate the practical details of fitting such distributions to sample data. Our hope is that the monograph will be useful to statisticians interested in mixture distributions and to research workers in other areas applying such distributions to their data. We would like to express our gratitude to Mrs Bertha Lakey for typing the manuscript. Institute oj Psychiatry University of London 1980 B.S. Everitt D.l Hand